258 XIV INTEGRATION IN LOCALLY COMPACT GROUPS If G is either compact or discrete, then we have mod(w) = 1 for every automorphism u of G. For it is clear that w(G) = G and u({e}) = {e}, and we may apply (14.3.6.2) with A = G and A = {e}, respectively. We deduce also from (14.3.6,2) that if u and v are two automorphisms of G, then (14.3.8) mod(w ° v) = mod(w) mod(v). (14.3.9) Ifuis any automorphism of the vector space R", then mod(w) = |det u\. Let U = (%ij) be the matrix of the automorphism u with respect to the canonical basis of R". Let Etj denote the n x n matrix which has the element in the (ij) place equal to 1 and all other elements equal to 0. If / ^j and 1 e R, put Then we have the following lemma: (14.3.9.1) Every invertible n x n matrix U is a product of matrices of the form Btj(X) and a matrix of the form In + (a~ l)Enn . Consider invertible matrices of the form /I 0 0 1 0 {lfB_fc 0 £2,n-h £i» \ 0 0 - 1 £n^h-lin-.h ' «.-*-!.. A A A e i: i I where /z^0;if/z = 7t-l, then .A" is an arbitrary invertible matrix. The proof is by induction on h. The matrix BU(X)X is obtained by adding A times the jth row to the fth row of X. If h = 0 we must have %nn ^ 0; hence if we mul- tiply on the left successively by the matrices Bin(-%~nl£in) for 1 g i <£ n - 1, we shall obtain the matrix / + (£nn - \)Enn, which proves the lemma in the case h = 0. Now suppose that the lemma has been proved for A = 0, 1, ..., k-\<n, and consider the case h = k. From the expansion for det(JT) we see Ann 7.4 that there must exist an element £i>n-k ^ 0 for some / such that n-k-^i^n. Premultiplying X by Bn((\ - £;> ' -*)C«-*) for some index j ^ i such that n-k^j^n, we see that we may assume that £/,-* = 1- Multiplying successively by the matrices ^r/(-^r)/J_fc) for r ^j9 we end uposed